Polynomial families of tautological classes on

We study classes Pg,T(α;β) on defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized with prescribed ramification over 0 and ∞. A comparison with classes Qg,T arising from sections of the universal Jacobian shows that the classes Pg,T(α;β) are polynomials in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.